How to compute the propogator for Chern-Simons on a torus?

Question

I'm looking to better understand Chern-Simons theory on a torus. We are given the action $$ S(\phi) = \int_E (\partial \phi)(\overline\partial \phi) + \frac{\lambda}{6}(\partial \phi)^3 $$ which yields (apparently) the propagator $$ P(z) = \frac{1}{4\pi}\wp(z;\tau) - \frac{1}{12}E_2(\tau) $$ (at least for $z \neq 0$)

What I would like to understand is the meaning of the terms in the action, as well as how the propagator is derived.

Classically, the action $$ \int_E (\partial \phi)(\overline\partial\phi) $$ yields the equation of motion $$ \partial\overline\partial\phi = 0 $$ or that $\phi(z, \overline z)$ is of the form $\phi_1(z) + \phi_2(\overline z)$. This then seems suggestive that the interaction term $(\partial \phi)^3$ tells us that we are only interested in holomorphic functions on $E$, which of course are going to be given by some combination of $\wp(z;\tau)$ and its derivatives. This seems reasonable, but other than an intuitive feel, I don't see why this is actually the case.

My questions are:

1. Is this a reasonable interpretation of the terms in the action?
2. Since my reasoning was based on classical ideas, how does this carry over to the quantum setting?
3. Given the above, how does one use this to derive that the propagator is as given above and not, say, $$ P(z) = \wp'(z)\big(\wp(z) + E_4(\tau)\big) $$ or something else similar?

I should probably specify that I am a mathematician, not a physicist.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose

Comments

To ensure that we are on the same page, which references are you mainly using for this?

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Qmechanic

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Well, not really any reference. The papers that I'm looking at are arxiv.org/abs/hep-th/9609022 and arxiv.org/abs/hep-th/9407176, say, but the genesis of my interest is Dijkgraaf's "Mirror Symmetry and Elliptic Curves".

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose

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I don't think this is related in any obvious way with Chern-Simons theory, could you explain why do you call it CS? Your references don't seem to involve CS at all either.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user fqq

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That's a good question. I'm not really sure---this is just something that I've seen it called. I know that formally the CS action $$\int tr(A \wedge dA + \frac{2}{3}A\wedge A \wedge A)$$ looks similar, but that's about all that I can say.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Simon Rose

Answer 1

Comments to the question (v1):

1. The theory that Ref. 1-3 are studying is not Chern-Simons theory, but an interacting scalar boson $\phi: S^1\times S^1\to \mathbb{R}$, living on the torus $S^1\times S^1$, i.e. a Riemann surface of genus one.

2. The propagator $\langle \phi(z)\phi(w)\rangle $ is here meant to be the free propagator. Thus we are considering a free scalar boson $\phi: S^1\times S^1\to \mathbb{R}$. This is a standard exercise, which can be found in many string theory textbooks, see e.g. Ref. 4-5.

3. In mathematical terms, the propagator $G(z,w)=\langle \phi(z)\phi(w)\rangle$ is just the Greens function for the Laplacian $\partial \bar{\partial}$ on the torus. In other words, we need to solve a double-periodic Dirichlet problem.

4. The double-periodic Greens function $G(z,w)$ can in principle be uniquely determined from knowledge of the pole structure. A perhaps more constructive type of derivation would involve the formula $$G(z,w)~=~\lim_{\varepsilon\searrow 0^+} \sum_n{}^{\prime} \frac{\psi_n(z)\psi^{\ast}_n(w)}{\lambda_n}e^{-\lambda_n\varepsilon },$$ where $\psi_n(z)$ denotes the eigenfunction for the eigenvalue $\lambda_n\geq 0$ of the Laplace operator $-\partial \bar{\partial}$, cf. e.g. Ref. 6. Here $\varepsilon>0$ is a regularization parameter. (Alternatively, one may use other regularization schemes.) The prime in the sum indicates that zeromodes should be excluded.

5. It turns out that the second derivative of the propagator $\partial_z\partial_w G(z,w)$ is a meromorphic function, i.e. it is holomorphic away from its poles. [Perhaps confusingly, the second derivative of the propagator $\partial_z\partial_w G(z,w)$ is referred to as 'the propagator' in Ref. 1 eq. (4.44).]

References:

1. R. Dijkgraaf, Chiral Deformations of Conformal Field Theories, Nucl. Phys. B493 (1997) 588, arXiv:hep-th/9609022.

2. R. Dijkgraaf, Mirror Symmetry and Elliptic Curves.

3. M.R. Douglas, Conformal Field Theory Techniques in Large N Yang-Mills Theory, arXiv:hep-th/9311130.

4. J. Polchinski, String Theory Vol. 1, 1998; Section 7.2.

5. E. Kiritsis, String Theory in a Nutshell, 2007; Section 4.18.3.

6. E. Cohen, H. Kluberg-Stern, H. Navelet, and R. Peschanski, Regulated propagator on the flat torus, CERN preprint.

This post imported from StackExchange Physics at 2014-05-04 11:23 (UCT), posted by SE-user Qmechanic